of the Motion of Incompressible Fluids. 131 



X being the distance of the point from a fixed horizontal plane. 

 Suppose 5 = — , z being an arbitrary function of 5, not neces- 

 sarily continuous, to which the transverse section of the tube 

 is always proportional, and u the value of q when z z= k. 



k du ku dz ds 



dt ~ % ' dT ~ z-^ ' ds' dt 



k du k*v^ _dz_ 



% ' dt ^ ' zds 



p sk du *^"' /i sdz\ /., , V 



Now as the shape of the tube and the function z are quite arbi- 

 trary, the motion which is taking place at any instant along its 

 axis, may be identical with the motion along a line drawn from 

 a point in the surface of fluid, maintained at a constant height 

 in any vessel whatever, to a point at the vena contracta, the 

 line being so drawn that it shall always be coincident with the 

 directions of the motions of the particles through which it 

 passes. Let therefore u = the velocity at the vena contracta, 

 k — the section of the stream at that point, z' = the area of 

 the upper surface of the fluid ; then will -^ = the vertical ve- 

 locity of a particle at any point of the surface, if the surface 

 retain its parallelism to the horizon. And if 9 = the Z which 

 the direction of the velocity of the particle makes with its 



surface, ^ " = its actual velocity. Suppose that when^ = P 

 the atmospheric pressure, x = 0, s = 0, z = 3*. 



n - P $k du A:' W^ / , sdz\ , *' u* 



$k du Ic'u'^ /. sdz\ , 



2 x" sin^ f 

 dz 



At the vena contracta z — k, p = F,'^ = 0, because ^ is a 

 minimum. Let s = <r. 



Hence o = g h - <r'^ ^"^ (l - ^,). For almost all 

 the points of the surface 9 will be very nearly 90° ; so that if 

 k be very small compared with z', ^,^_,.^^^ may be neglected. 

 And because the velocity of issuing must very soon be uni- 

 form, after a very short time from tne commencement of mo- 



du 



tion — = 



dt 



Therefore u^ = 2gh. 



It thus appears that independently of the motions of the 



panicles in the iiiltrior of the vessel, (for tlicy may be any 



S 2 whatever 



